On Thursday, September 18, 2025 at 11:28:06 PM UTC-6 Brent Meeker wrote:
On 9/18/2025 8:15 AM, Alan Grayson wrote: On Thursday, September 18, 2025 at 7:23:39 AM UTC-6 John Clark wrote: On Mon, Sep 15, 2025 at 4:16 AM Alan Grayson <[email protected]> wrote: *> If you claim there is no acceleration in GR, I'd like to know how acceleration is defined, and how it can be calculated to be zero. I tried using the 2nd derivative of dS, but it doesn't yield zero. AG* *That's how acceleration is determined in Newtonian physics but that's not good enough for General Relativity. Newton assumed that time proceeded at a constant rate for everybody, but Einstein knew that wasn't true, you have to consider your motion not just through space but also your motion through time. **And Einstein realized that not only is it possible to accelerate through space it is also possible to accelerate through time. * *In General Relativity your "position" isn’t just defined by (x,y,z), but by (t,x,y,z), it's called your "4-position" and mathematically is expressed as a 4-vector, or as a rank-1 tensor. The mathematical equation that defines it uses Christoffel symbols that encode the 4-D non-Euclidean spacetime curvature you get in a gravitational field.* *Please elaborate on the role of the Christoffel symbols that enclode the spacetime curvative in a gravitational field. What does "encode" mean in this context? AG* * It qualifies as being a tensor because when you change coordinates, that is to say when you switch from one observer's frame of reference to another, although its individual components might change, the tensor as a whole does not change.* *Can a tensor be defined as a mathematical entitity which is invariant under a coordinate transformation. * Not, invariant but covariant. *What is the form of the entity that guarantees that invariance? * It's not the form. For example you could write down the matrix defining a 2nd order tensor in some particular coordinate system and then it would be defined in every other coordinate system by its transformation to that system. So it's not the same matrix in the new system, but it is the same physical object...that's why it's call covariant, not invariant. This often done in relative theory by choosing the coordinate system in which it is relatively easy to write down the tensor and then transforming that to the system of interest. *Must the entity being measured in different frames have a particular form, say as a matrix, for the invariance to make sense? AG * *According to Einstein a body that is not experiencing a force always follows the straightest possible line, and in curved 4-D spacetime that would be a geodesic. You can use your 4-position to calculate your "proper acceleration", the acceleration you feel and that registers on an accelerometer. And you can also use it to calculate your "coordinate acceleration", the change of spatial coordinate you get when you change from one coordinate system to another, or to put it another way, when you change from one frame of reference to another.* *Proper acceleration is invariant, every observer agrees whether or not you feel acceleration, (emphasis added, AG)* *Brent says, I think, that proper accelation will not be felt. True or not? AG * No. Brent says no acceleration is felt when proper acceleration is zero. Even though the Newtonian acceleration (the coordinate 3-acceleration) is non-zero, e.g. in orbiting. I thought you had bought a text book so you could read this stuff. Brent *No book will tell me if Clark misstated what Brent claimed. Is it on Amazon? AG* *Is this Newtonian acceleration? AG * *Einstein says that everything is always moving through spacetime at the speed of light, when you're just sitting under a tree and not moving all your speed is in the time direction, but when you get up and start walking a small part of your velocity vector is in the spatial direction and so your speed through the time dimension is reduced slightly. If you’re in a spaceship orbiting Earth then the space-time curvature (mathematically expressed as Christoffel symbols) bends your path through both 3-D space and 40 spacetime, but you haven't fired your rockets so your proper acceleration is zero and you feel weightless.* * John K Clark See what's on my new list at Extropolis <https://groups.google.com/g/extropolis>* -- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to [email protected]. To view this discussion visit https://groups.google.com/d/msgid/everything-list/38e1fe14-411b-40ee-a4dd-c15e2bd9c4c2n%40googlegroups.com.
